Planar Drawing Research Articles

Testing Planarity of Partially Embedded Graphs

We study the following problem: given a planar graph G and a planar drawing (embedding) of a subgraph of G , can such a drawing be extended to a planar drawing of the entire graph G ? This problem fits the paradigm of extending a partial solution for a problem to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes an otherwise easy problem hard, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmas, which show that the planarity of partially embedded graphs exhibits the ‘TONCAS’ behavior “the obvious necessary conditions for planarity are also sufficient.” These conditions are expressed in terms of the interplay between (1) the rotation system and containment relationships between cycles and (2) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we make our algorithm run in linear time. Finally, we consider several generalizations of the problem, such as minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. We also apply our algorithm to the simultaneous graph drawing problem Simultaneous Embedding with Fixed Edges (Sefe) . There we obtain a linear-time algorithm for the case that one of the input graphs or the common graph has a fixed planar embedding.

Disconnectivity and relative positions in simultaneous embeddings

For two planar graphs G①=(V①,E①) and G②=(V②,E②) sharing a common subgraph G=G①∩G② the problem Simultaneous Embedding with Fixed Edges (SEFE) asks whether they admit planar drawings such that the common graph is drawn the same. Previous algorithms only work for cases where G is connected, and hence do not need to handle relative positions of connected components. We consider the problem where G, G① and G② are not necessarily connected.First, we show that a general instance of SEFE can be reduced in linear time to an equivalent instance where V①=V② and G① and G② are connected. Second, for the case where G consists of disjoint cycles, we introduce the CC-tree which represents all embeddings of G that extend to planar embeddings of G①. We show that CC-trees can be computed in linear time, and that their intersection is again a CC-tree. This yields a linear-time algorithm for SEFE if all k input graphs (possibly k>2) pairwise share the same set of disjoint cycles. These results, including the CC-tree, extend to the case where G consists of arbitrary connected components, each with a fixed planar embedding on the sphere. Then the running time is O(n2).

Constrained Point Set Embedding of a Balanced Binary Tree

Given an undirected planar graph G with n vertices and a set S of n points inside a simple polygon P, a point-set embedding of G on S is a planar drawing of G such that each vertex is mapped to a distinct point of S and the edges are polygonal chains surrounded by P. A special case of the embedding problem is that in which G is a balanced binary tree. In this paper, we present a new algorithm for embedding an n-vertex balanced binary tree BBT on a set S of n points inside a simple m-gon P in O(m4/3+ϵ + nlog2 n + m log n) time with at most O(m) bends per edge.

Upward planar graphs and their duals

We consider upward planar drawings of directed graphs in the plane (UP), and on standing (SUP) and rolling cylinders (RUP). In the plane and on the standing cylinder the edge curves are monotonically increasing in y-direction. On the rolling cylinder they wind unidirectionally around the cylinder. There is a strict hierarchy of classes of upward planar graphs: UP⊂SUP⊂RUP.In this paper, we show that rolling and standing cylinders switch roles when considering an upward planar graph and its dual. In particular, we prove that a strongly connected graph is RUP if and only if its dual is a SUP dipole. A dipole is an acyclic graph with a single source and a single sink. All RUP graphs are characterized in terms of their duals using generalized dipoles. Moreover, we obtain a characterization of the primals and duals of wSUP graphs which are upward planar graphs on the standing cylinder and allow for horizontal edge curves.

Plane 3-Trees: Embeddability and Approximation

We give an $O(n\log ^3 n)$-time linear-space algorithm that, given a plane 3-tree $G$ with $n$ vertices and a set $S$ of $n$ points in the plane, determines whether $G$ has a point-set embedding on $S$ (i.e., a planar straight-line drawing of $G$ where each vertex is mapped to a distinct point of $S$), improving the $O(n^{4/3+\varepsilon})$-time $O(n^{4/3})$-space algorithm of Moosa and Rahman [Lecture Notes in Comput. Sci. 6842, Springer, New York, 2011, pp. 204--212]. Given an arbitrary plane graph $G$ and a point set $S$, Kaufmann and Wiese [J. Graph Algorithms Appl., 6 (2002), pp. 115--129] gave an algorithm to compute 2-bend point-set embeddings of $G$ on $S$. Later, Di Giacomo and Liotta [Lecture Notes in Comput. Sci. 5942, Springer, New York, 2010, pp. 35--46] showed how such a drawing can be computed using $O(W^3)$ area, where $W$ is the length of the longest edge of the bounding box of $S$. Their algorithm uses $O(W^3)$ area even when the input graphs are restricted to plane 3-trees. We introduce new techniques for computing $2$-bend point-set embeddings of plane 3-trees that take only $O(W^2)$ area. We also give approximation algorithms for point-set embeddings of plane $3$-trees. Our results on 2-bend point-set embeddings and approximate point-set embeddings hold for partial plane $3$-trees (e.g., series-parallel graphs and Halin graphs).

Various heuristic algorithms to minimise the two-page crossing numbers of graphs

Abstract We propose several new heuristics for the twopage book crossing problem, which are based on recent algorithms for the corresponding one-page problem. Especially, the neural network model for edge allocation is combined for the first time with various one-page algorithms. We investigate the performance of the new heuristics by testing them on various benchmark test suites. It is found out that the new heuristics outperform the previously known heuristics and produce good approximations of the planar crossing number for severalwell-known graph families. We conjecture that the optimal two-page drawing of a graph represents the planar drawing of the graph.

Planar embedding of trees on point sets without the general position assumption

The problem of point-set embedding of a planar graph $G$ on a point set $P$ in the plane is defined as finding a straight-line planar drawing of $G$ such that the nodes of $G$ are mapped one to one on the points of $P$. Previous works in this area mostly assume that the points of $P$ are in general position, i.e. $P$ does not contain any three collinear points. However, in most of the real applications we cannot assume the general position assumption. In this paper, we show that deciding the point-set embeddability of trees without the general position assumption is NP-complete. Then we introduce an algorithm for point-set embedding of $n$-node binary trees with at most $\frac{n}{3}$ total bends on any point set. We also give some results when the problem is limited to degree-constrained trees and point sets having constant number of collinear points.

Computing upward topological book embeddings of upward planar digraphs

We describe a unified approach for studying book, point-set, and simultaneous embeddability problems of upward planar digraphs. The approach is based on a linear time strategy to compute an upward planar drawing of an upward planar digraph such that all vertices are collinear. Besides having impact in relevant application domains of graph drawing and computational geometry, the presented results open new research directions in the area of upward planarity with constraints of the positions of the vertices.

Algorithms for Drawing Planar p-petal Graphs

In this paper, we present some algorithms for drawing planar p-petal graphs. Algorithms for layered drawing, orthogonal drawing, proximity drawing and visibility drawing are also presented.

Study on the Correlation between Drawing Education and Special Cognitive Ability Evaluation Based on the MCT

Use the international universal theory of evaluation of special cognitive ability with Mental Cutting Test (MCT) to test the students of Science and Technology University. Compare the MCT average scores of fore-and-aft drawing education, it shows that the effect of descriptive geometry teaching is positive correlative with the scores of MCT, the students improved their special cognitive ability from planar drawing to three-dimensional drawing significantly. The students who got lower MCT scores before taking drawing courses improved their MCT scores obviously. The effect of mechanical drawing teaching is feeble correlative with the scores of MCT; it shows feeble improvement in students’ special cognitive ability.

A note on 1-planar graphs

A graph is 1-planar if it can be drawn in the plane such that each of its edges is crossed at most once. We prove a conjecture of Czap and Hudák (2013) stating that the edge set of every 1-planar graph can be decomposed into a planar graph and a forest. We also provide simple proofs for the following recent results: (i) an n-vertex graph that admits a 1-planar drawing with straight-line edges has at most 4n−9 edges (Didimo, 2013); and (ii) every drawing of a maximally dense right angle crossing graph is 1-planar (Eades and Liotta, 2013).

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.

On the Upward Planarity of Mixed Plane Graphs

A mixed plane graph is a plane graph whose edge set is partitioned into a set of directed edges and a set of undirected edges. An orientation of a mixed plane graph G is an assignment of directions to the undirected edges of G resulting in a directed plane graph $\vec G$ . In this paper, we study the computational complexity of testing whether a given mixed plane graph G is upward planar, i.e., whether it admits an orientation resulting in a directed plane graph G such that G admits a planar drawing in which each edge is represented by a curve monotonically increasing in the y-direction according to its orientation. Our contribution is threefold. First, we show that the upward planarity testing problem is solvable in cubic time for mixed outerplane graphs. Second, we show that the problem of testing the upward planarity of mixed plane graphs reduces in quadratic time to the problem of testing the upward planarity of mixed plane triangulations. Third, we exhibit linear-time testing algorithms for two classes of mixed plane triangulations, namely mixed plane 3-trees and mixed plane triangulations in which the undirected edges induce a forest.

Universal Point Sets for Drawing Planar Graphs with Circular Arcs

We prove that there exists a set S of n points in the plane such that every n-vertex planar graph G admits a plane drawing in which every vertex of G is placed on a distinct point of S and every edge of G is drawn as a circular arc.

Topological morphing of planar graphs

In this paper we analyze the relationships among different planar embeddings of the same graph and study how two planar embeddings can be morphed one into the other with the minimum number of elementary changes, while preserving the mental map of the user. We call this problem Topological Morphing, in analogy with the well-known Geometric Morphing problem, in which it is studied how two planar drawings can be morphed one into the other with the minimum number of elementary changes.First, we define two operations, called flip and skip, describing natural transformations of a graph embedding that preserve the mental map of the user. Then, we study the problem of performing a morph while minimizing the number of these operations. We show that the Topological Morphing problem is NP-hard for biconnected planar graphs, we give polynomial-time algorithms for some restricted versions of the problem, and, based on such polynomial-time algorithms, we give a fixed-parameter tractable algorithm for the general case.

Upward planar drawings on the standing and the rolling cylinders

We consider directed graphs with an upward planar drawing on the standing and the rolling cylinders. These graphs properly extend the upward planar graphs in the plane. The drawings allow complex curves for the edges with many zig-zags and windings around the cylinder.We show that the drawings can be simplified to polyline drawings with geodesics as straight segments, at most two bends per edge, and vertices and bends on a grid of at most cubic size. On the standing cylinder the edges do not wind and they wind at most once around the rolling cylinder, where single windings can be unavoidable. The simplifications can be computed efficiently in O(τn3) time, where τ is the cost of computing the point of intersection of a curve and a horizontal line through a vertex.Moreover, we provide a complete classification of the classes of (regular) upward planar drawable graphs on the plane, the sphere, and the standing and rolling cylinders including their strict and weak variants. Strict is upward in two dimensions and weak allows horizontal lines. For upward planarity the standing and the rolling cylinders coincide in the strict case, rolling dominates standing in the regular case, and there is an incomparability in the weak case. The respective classes of graphs range properly between the upward planar and the quasi-upward planar graphs, and they have an NP-hard recognition problem.

On Drawings and Decompositions of 1-Planar Graphs

A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. We show that every 1-planar drawing of any 1-planar graph on $n$ vertices has at most $n-2$ crossings; moreover, this bound is tight. By this novel necessary condition for 1-planarity, we characterize the 1-planarity of Cartesian product $K_m\times P_n$. Based on this condition, we also derive an upper bound on the number of edges of bipartite 1-planar graphs, and we show that each subgraph of an optimal 1-planar graph (i.e., a 1-planar graph with $n$ vertices and $4n-8$ edges) can be decomposed into a planar graph and a forest.

Program Fullerene: A software package for constructing and analyzing structures of regular fullerenes

Fullerene (Version 4.4) is a general purpose open-source program that can generate any fullerene isomer, perform topological and graph theoretical analysis, as well as calculate a number of physical and chemical properties. The program creates symmetric planar drawings of the fullerene graph and generates accurate molecular 3D geometries by way of force-field optimization, serving as a good starting point for further quantum theoretical treatments. It includes a number of fullerene-to-fullerene transformations, such as Goldberg-Coxeter transforms, Stone-Wales transforms, Endo-Kroto, Yoshida-Fowler, and Brinkmann-Fowler vertex insertions. The program is written in standard Fortran and C++ and can easily be installed in a Linux or UNIX environment.

SIMULTANEOUS EMBEDDING OF EMBEDDED PLANAR GRAPHS

A simultaneous embedding with fixed edges (SEFE) of a set of k planar graphs G1,…,Gk on the same set of vertices is a set of k planar drawings of G1,…,Gk, respectively, such that each vertex is placed on the same point in all the drawings and each edge is represented by the same Jordan curve in the drawings of all the graphs it belongs to. A simultaneous geometric embedding (SGE) is a SEFE in which the edges are represented by straight-line segments. Given k planar graphs G1,…,Gk, deciding whether they admit a SEFE and whether they admit an SGE are NP-hard problems, for k ≥ 3 and for k ≥ 2, respectively. In this paper we consider the complexity of SEFE and of SGE when the graphs G1,…,Gk have a fixed planar embedding. In sharp contrast with the NP-hardness of SEFE for three non-embedded planar graphs, we show that SEFE is quadratic-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G1,…,Gk, deciding whether a SEFE of G1,…,Gk exists and deciding whether an SGE of G1,…,Gk exists are NP-hard problems, for k ≥ 14 and k ≥ 13, respectively.

Density of straight-line 1-planar graph drawings

A 1-planar drawing of a graph is such that each edge is crossed at most once. In 1997, Pach and Tóth showed that any 1-planar drawing with n vertices has at most 4n−8 edges and that this bound is tight for n⩾12. We show that, in fact, 1-planar drawings with n vertices have at most 4n−9 edges, if we require that the edges are straight-line segments. We also prove that this bound is tight for infinitely many values of n⩾8. Furthermore, we investigate the density of 1-planar straight-line drawings with additional constraints on the vertex positions. We show that 1-planar drawings of bipartite graphs whose vertices lie on two distinct horizontal layers have at most 1.5n−2 edges, and we prove that 1-planar drawings such that all vertices lie on a circumference have at most 2.5n−4 edges; both these bounds are also tight.